All Trigonometric Formulas

All Trigonometric Formulas for Government Exams

A comprehensive, colorful guide with formulas, graphs, quadrants, and applications.

Trigonometric Ratios

Ratio	Formula	Description
sin θ	Perpendicular / Hypotenuse	Opposite side over longest side
cos θ	Base / Hypotenuse	Adjacent side over longest side
tan θ	Perpendicular / Base	Opposite over adjacent
cot θ	Base / Perpendicular	Adjacent over opposite
sec θ	Hypotenuse / Base	Longest side over adjacent
csc θ	Hypotenuse / Perpendicular	Longest side over opposite

Pythagorean Identities

Identity	Formula
Basic	sin²θ + cos²θ = 1
Tangent	1 + tan²θ = sec²θ
Cotangent	1 + cot²θ = csc²θ
Rearranged	sin²θ = 1 - cos²θ
Rearranged	cos²θ = 1 - sin²θ

Reciprocal and Quotient Identities

Type	Formula
Reciprocal	sin θ = 1 / csc θ
Reciprocal	cos θ = 1 / sec θ
Reciprocal	tan θ = 1 / cot θ
Quotient	tan θ = sin θ / cos θ
Quotient	cot θ = cos θ / sin θ

Angle Sum and Difference Identities

Function	Sum Formula	Difference Formula
sin	sin(α + β) = sin α cos β + cos α sin β	sin(α - β) = sin α cos β - cos α sin β
cos	cos(α + β) = cos α cos β - sin α sin β	cos(α - β) = cos α cos β + sin α sin β
tan	tan(α + β) = (tan α + tan β) / (1 - tan α tan β)	tan(α - β) = (tan α - tan β) / (1 + tan α tan β)

Double and Half Angle Identities

Type	Formula
Double (sin 2θ)	2 sin θ cos θ
Double (cos 2θ)	cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
Double (tan 2θ)	(2 tan θ) / (1 - tan²θ)
Half (sin²(θ/2))	(1 - cos θ) / 2
Half (cos²(θ/2))	(1 + cos θ) / 2
Half (tan(θ/2))	(1 - cos θ) / sin θ = sin θ / (1 + cos θ)

Product-to-Sum and Sum-to-Product Identities

Type	Formula
Product-to-Sum	sin α cos β = (1/2)[sin(α + β) + sin(α - β)]
Product-to-Sum	cos α cos β = (1/2)[cos(α + β) + cos(α - β)]
Sum-to-Product	sin α + sin β = 2 sin((α + β)/2) cos((α - β)/2)
Sum-to-Product	cos α + cos β = 2 cos((α + β)/2) cos((α - β)/2)

Quadrant Signs

Quadrant	sin θ	cos θ	tan θ	cot θ	sec θ	csc θ
I (0°-90°)	+	+	+	+	+	+
II (90°-180°)	+	-	-	-	-	+
III (180°-270°)	-	-	+	+	-	-
IV (270°-360°)	-	+	-	-	+	-

Mnemonic: "All Students Take Calculus" (All in I, Sin in II, Tan in III, Cos in IV).

Standard Angle Values

Angle	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	Undefined

Inverse Trigonometric Identities

Identity	Formula
Complementary	sin⁻¹ x + cos⁻¹ x = π/2
Negative	sin⁻¹ (-x) = -sin⁻¹ x
Tangent Sum	tan⁻¹ x + tan⁻¹ y = tan⁻¹ ((x + y)/(1 - xy))

Triangle Formulas

Type	Formula
Law of Sines	a/sin A = b/sin B = c/sin C
Law of Cosines	c² = a² + b² - 2ab cos C
Area (Sine)	Area = (1/2)ab sin C
Heron’s Formula	Area = √[s(s-a)(s-b)(s-c)], s = (a+b+c)/2

Applications (Heights, Distances, Algebra)

Application	Formula/Example
Height Calculation	Height = Distance × tan θ
Algebraic Simplification	Simplify: sin²θ + cos²θ → 1
Equation Solving	Solve: 2sin θ = 1 → sin θ = 1/2, θ = 30°

Graphs of Trigonometric Functions

1. Sine Function (sin θ)

Description: Periodic wave, period 360°, amplitude 1, oscillates between -1 and 1. Crosses x-axis at 0°, 180°, 360°.

[Graph Placeholder: Insert sin θ graph or view at Desmos: y = sin(x)]

2. Cosine Function (cos θ)

Description: Similar to sine, shifted left by 90°. Period 360°, amplitude 1, starts at (0,1).

[Graph Placeholder: Insert cos θ graph or view at Desmos: y = cos(x)]

3. Tangent Function (tan θ)

Description: Period 180°, undefined at 90°, 270°. Crosses x-axis at 0°, 180°.

[Graph Placeholder: Insert tan θ graph or view at Desmos: y = tan(x)]

Quadrant Diagram

Description: A circle divided into four quadrants:
- I: (x > 0, y > 0), all positive.
- II: (x < 0, y > 0), sin/csc positive.
- III: (x < 0, y < 0), tan/cot positive.
- IV: (x > 0, y < 0), cos/sec positive.

[Diagram Placeholder: Insert unit circle with quadrants or view at Desmos: Unit Circle]